Compact representation of multivariate probability distributions
Probabilistic graphical models for a set of variables \(\{ X_1, X_2, \ldots, X_n \}\) characterized by
- a graphical structure, directed and acyclic, whose nodes are the variables
- a probability model for each node describing the relationship with its parents
- edges encode conditional independencies (any variable is conditionally independent of its non-descendant given its parents)
$P(X_1)P(X_2)P(X_3 \vert X_1, X_2)P(X_4 \vert X_2, X_3)$ e.g. $X_4 \perp\!\!\!\perp X_1 \vert (X_2, X_3)$
\(\{ X_1, X_2, \ldots, X_n \} \thicksim P(X_1, X_2, \ldots, X_n) = \prod_{i=1}^n P(X_i \vert {\textbf{Pa}_i})\)
sample of 10,000 DAGs